In the SPDE formulation of the problem, the latent field \(\mathbf {\omega }_t\)  is discretized on a mesh with \(D\)  vertices as described in equation (REF ). Denoting by \(\mathbf {\omega }_t^D\)  the vector of values at the nodes of the mesh at time \(t\) , then \(\mathbf {\omega }_t^D \sim N(\mathbf {0}, \mathbf {Q}_s^{-1})\)  where \(\mathbf {Q}_s\)  is the precision matrix whose values are computed using (REF ). Details of the computation can be found in [1]}. \(\mathbf {Q}_s\)  is a sparse precision matrix whose matrix inverse provides a good approximation of the variance-covariance matrix with elements computed using the Matèrn covariance function. This gives
\(\mathbf {\xi }_t^D = \varsigma \mathbf {\xi }_{t-1}^D + \mathbf {\omega }_t^D, \;\;\; \mathbf {\omega }_t^D \sim N(\mathbf {0},\textbf {Q}_s^{-1}), \;\;\; t=1,\ldots ,T,\) 
